[Paper Review] Hypergraph Dissimilarity Measures

Amit Surana, Can Chen+1 more|arXiv (Cornell University)|Jun 15, 2021

Tensor decomposition and applications51 references4 citations

TL;DR

This paper proposes two novel approaches for hypergraph dissimilarity measurement: an indirect method that transforms hypergraphs into graphs using clique or star expansion to leverage existing graph dissimilarity measures, and a direct method that uses tensor algebra—specifically adjacency and Laplacian tensors with eigenvalues and singular values—to intrinsically capture multi-way relationships. The key contribution is a scalable, multi-scale framework for comparing hypergraphs, validated on synthetic and biological datasets with statistically significant results.

ABSTRACT

In this paper, we propose two novel approaches for hypergraph comparison. The first approach transforms the hypergraph into a graph representation for use of standard graph dissimilarity measures. The second approach exploits the mathematics of tensors to intrinsically capture multi-way relations. For each approach, we present measures that assess hypergraph dissimilarity at a specific scale or provide a more holistic multi-scale comparison. We test these measures on synthetic hypergraphs and apply them to biological datasets.

Motivation & Objective

• To address the lack of standardized hypergraph dissimilarity measures (HDMs) in contrast to the well-established graph dissimilarity measures (GDMs).
• To develop scalable, multi-scale HDMs that can assess structural differences at local, mesoscopic, and global levels.
• To enable comparison of real-world biological hypergraphs, such as chromosomal contact maps from fibroblasts and B lymphocytes.
• To evaluate the trade-offs between indirect (graph-based) and direct (tensor-based) HDM approaches in terms of computational efficiency and structural fidelity.

Proposed method

• Transform hypergraphs into graphs via clique expansion, where each hyperedge becomes a complete clique, preserving pairwise relationships.
• Transform hypergraphs into graphs via star expansion, where each hyperedge is represented as a star topology centered on a hyperedge node.
• Apply standard graph dissimilarity measures (e.g., Hamming, spectral, deltaCon, centrality) to the transformed graphs to derive indirect HDMs.
• Represent hypergraphs using adjacency and Laplacian tensors to preserve multi-way interactions intrinsically.
• Utilize tensor algebraic concepts—tensor eigenvalues, eigenvectors, and higher-order singular values—to define direct HDMs.
• Use permutation tests to assess statistical significance of observed dissimilarities between hypergraphs.

Experimental results

Research questions

• RQ1 Can hypergraph comparison be effectively achieved by transforming hypergraphs into graphs and applying existing graph dissimilarity measures?
• RQ2 Can tensor-based representations of hypergraphs enable more accurate, intrinsic multi-way relationship comparison than graph-based projections?
• RQ3 How do indirect and direct HDMs perform in distinguishing between different hypergraph topologies and real biological networks?
• RQ4 What are the trade-offs between computational efficiency and structural fidelity in indirect versus direct HDM approaches?

Key findings

• The indirect HDMs based on clique expansion detected statistically significant differences between fibroblast (FB) and B lymphocyte (GM) chromosomal contact maps at all scales, with p-values < 10^-2.
• Star expansion-based indirect HDMs showed significant differences at local and global scales, but similarity at the mesoscopic scale, indicating scale-dependent structural variation.
• The deltaCon measure applied via clique expansion yielded a p-value of 4.7×10^-7, indicating highly significant dissimilarity between FB and GM hypergraphs.
• The direct HDMs, based on tensor algebra, offer a lossless representation of multi-way relationships but face computational challenges with large hypergraphs or high hyperedge cardinality.
• Permutation tests confirmed that observed dissimilarities between FB and GM are statistically significant across multiple indirect HDMs, validating their discriminative power.